The purpose of this study is to find out how to increase the potential for achieving an "operational cure" from chronic myeloid leukemia. An "operational cure" is a state in which a person does not require further treatment, although there may be some remaining cancer cells. Patients would normally remain on a TK inhibitor indefinitely within a standard of care setting for chronic myeloid leukemia. Within this clinical trial, patients will discontinue their TK inhibitor prematurely. If any signs of progression are identified, dasatinib will be introduced.